The number of emails a professor receives from students each

The number of emails a professor receives from students each hour on the day before an exam is a Poisson random variable. Recall that a Poisson random variable X is the number of events that occur in some interval and that it has the following probability mass function:

Suppose that instead of being interested in the number of events (a discrete random variable), we are interested in the size of the interval between successive events (a continuous random variable). Let Y denote the length of time from any starting point until an event occurs.

1. Assume that emails arrive at the rate of ? = 0.1 emails per minute. What is the probability that the time between successive emails is at most 40 minutes? [Hint: If two events are successive then no other events occurred in the interval between them.]

2.Given a Poisson random variable X with rate parameter ? per unit interval, what is the probability that the interval between successive events, random variable Y, is less than or equal to y?

3. What is the probability density function that corresponds to the cumulative distribution function you derived in question 2?

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Solution

The number of emails a professor receives from students each hour on the day before an exam is a Poisson random variable.

A random variable X is the number of events that occur in some interval.

Given that, X ~ Poison ( µ ).

Let Y denote the length of time from any starting point until an event occurs.

Here in random variable Y given the interval and we cannot calculate probabilities of the interval in discrete distribution.

So we go for Exponential distribution that is continous distribution.

Y ~ Exp( µ )

The probability distribution function of Y is,

f ( Y ) = µ * e^-µY (y is from 0 to infinity)

a) Given that µ = 0.1 emails per minute.

What is the probability that the time between successive emails is at most 40 minutes?

P(Y 40) = 1 - e^-µY

= 1 - e^(-0.1*40) = 1 - 0.01832 = 0.9817

b) Given a Poisson random variable X with rate parameter ? per unit interval, what is the probability that the interval between successive events, random variable Y, is less than or equal to y?

P( Y y ) = f(Y) dY (integral is from o to y)

= µ * e^-µY dY

= µ e^-µY dY

= µ [ e^-µY / -µ ] (integral is from o to y)

µ get cansle,

P( Y y ) = 1 - e^-µY   

c) What is the probability density function that corresponds to the cumulative distribution function you derived in question 2?

We can calculate in b) part P( Y y ) is nothing but the cumulative distribution function.

We want to write the probability distribution function of y is,

f(y) = µ * e ^-µy